Related papers: A Dirichlet type problem for non-pluripolar comple…
We give two applications of the the duality between the complex Homogeneous Monge-Amp\`ere Equation (HMAE) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for…
We prove the long time existence and uniqueness of solution to a parabolic quaternionic Monge-Amp\`{e}re type equation on a compact hyperK\"{a}hler manifold. We also show that after normalization, the solution converges smoothly to the…
In this note we study a complex Monge-Amp\`{e}re type equation of the form (dd^cu)^n = \frac {ke^{-u}dV}{\int e^{-u}dV}\,
Let $\M$ be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into $\M$ with prescribed singularities along a…
We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…
In this paper, we consider the Monge-Amp\`{e}re type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence…
We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we…
We consider a variational problem with boundary singularity and Dirichlet condition. We give a blow-up analysis for sequences of solutions of an equation with exponential nonlinearity. Also, we derive a compactness criterion under some…
We prove a number of results related to the size and propagation of boundary pluripolar sets, the exceptional sets for the Dirichlet problem for the complex Monge--Amp\`ere equation. We extend Stout's result that peak sets on strictly…
In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Amp\`ere equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates…
In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Amp\`ere equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.
We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Amp\`ere equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the…
We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Amp\`ere equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on…
We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $\Omega$ in $\mathbb{R}^n$, $n \ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space…
We construct convex functions on $\mathbb{R}^3$ and $\mathbb{R}^4$ that are smooth solutions to the Monge-Amp\`{e}re equation $\det D^2u = 1$ away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a…
We introduce a parabolic analogue of the elliptic split-type Monge-Amp\`ere equation developed by Fang and the author, extending Streets' twisted Monge-Amp\`ere equation. The resulting equation is fully nonlinear and non-concave. We prove…
Given a psh function $\varphi\in\mathcal{E}(\Omega)$ and a smooth, bounded $\theta\geq 0$, it is known that one can solve the Monge-Amp\`{e}re equation $\mathrm{MA}(\varphi_\theta)=\theta^n\mathrm{MA}(\varphi)$, with some form of Dirichlet…
A PDE proof is provided for the sharp $L^\infty$ estimates for the complex Monge-Amp\`ere equation which had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics.…
We study global convex solutions of the Monge-Amp\`ere equation \[ \det D^2 u = \mu \quad \text{in } \mathbb{R}^n, \] where $\mu \not\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We prove a…
In this paper we study the following eigenvalue boundary value problem for Monge-Amp\`{e}re equations: {equation} \{{array}{l} \det(D^2u)=\lambda^N f(-u)\,\, \text{in}\,\, \Omega, u=0,\,\text{on}\,\, \partial \Omega. {array}. {equation} We…